Generalized Resultant Theorem

Vardulakis, Α. I. G.; Stoyle, P. N. R.

doi:10.1093/imamat/22.3.331

Cited by 29 publications

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“…the Sylvester matrix of f and g. If the resulting triangularized matrix is arranged as a right upper triangular matrix, the bottom most non-zero row corresponds to a gcd of the polynomials f and g. As a first step we generalize this method to the case of r univariate polynomials with r ≥ 2. Such generalizations have been done before (see [5], [2]). Those, however, resulted in bigger matrices.…”

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confidence: 82%

Gröbner bases and generalized sylvester matrices

Wiesinger-Widi

2011

ACM Commun. Comput. Algebra

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In his PhD thesis [1], Buchberger introduced the notion of Gröbner bases and gave the first algorithm for computing them. Since then, extensive research has been done in order to reduce the complexity of the computation. But nevertheless, even for small examples the computation sometimes does not terminate in reasonable time.There are basically two approaches for computing a Gröbner basis. The first is the one pursued by the Buchberger algorithm: We start from the initial set F , execute certain reduction steps (consisting of multiplication of polynomials by terms -called shifts -and subtraction of polynomials) and due to Buchberger's theorem, which says that the computation is finished if all the s-polynomials reduce to zero, we know that after finitely many iterations of this procedure we obtain a Gröbner basis of the ideal generated by F . The second approach is to start from F , execute certain shifts of the initial polynomials in F , arrange them as rows in a matrix, triangularize this matrix and from the resulting matrix extract a Gröbner basis.In project DK1 of the Doctoral Program, which was proposed by Buchberger, we pursue the second approach and seek to improve the theory in order to speed up the Gröbner bases computation. This approach has been studied a couple of times in the past, but never thoroughly. The immediate question is: Does there exist a finite set of shifts such that a triangularization of the matrix built by these shifts yields a Gröbner basis and, if so, how can we construct these shifts? We give first results in answering this question. In the following, let K be a field.In the univariate case, Gröbner bases computation specializes to gcd computation. In [3] (see also [4] for a good overview on this topic), Habicht establishes a connection between the computation of polynomial remainder sequences and linear algebra. More specifically, the problem of finding a gcd of two polynomials f, g ∈ K[x] with degrees m and n, respectively, where m ≥ n, can be solved by triangularizing the matrix

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confidence: 82%

Gröbner bases and generalized sylvester matrices

Wiesinger-Widi

2011

ACM Commun. Comput. Algebra

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“…Differently, it is shown below that for p > 2 the matrix R is not a resultant matrix as it can be singular even if the polynomials a(rjz), 0 ≤ j ≤ p − 1 have no common factors. Extensions of the classical resultant theorem for more than two polynomials (p > 2) are established in [1,13], but they employ rectangular generalized resultant matrices. πi z) and a(e 4 3 πi z) are pairwise relatively prime, it is easily found that the corresponding 9 × 9 matrix R is singular.…”

Section: The Polynomial Approach and Its Matrix Counterparts For Genementioning

confidence: 99%

Solving Bezout-like polynomial equations for the design of interpolatory subdivision schemes

Conti

Gemignani

Romani

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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Subdivision schemes are nowadays customary in curve and surface modeling. In this paper the problem of designing interpolatory subdivision schemes is considered. The idea is to modify a given approximating subdivision scheme just enough to satisfy the interpolation requirement. This leads to the solution of a generalized Bezout polynomial equation possibly involving more than two polynomials. By exploiting the matrix counterpart of this equation it is shown that small-degree solutions can be generally found by inverting an associated structured matrix of Toeplitz-like form.

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“…The above suggests that the resultant set may be used for the evaluation of GCD and the resultant properties expressing these links are summarised below [10,12,13]. The Sylvester resultant result stated above is central in establishing a number of important computational procedures for the GCD of many polynomials.…”

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confidence: 99%

“…The classical approaches for the study of coprimeness and determination of the GCD makes use of the Sylvester resultant which in the case of many polynomials is defined as shown below [12,13] …”

Section: Introductionmentioning

confidence: 99%

Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation

Karcanias

Fatouros

Mitrouli

et al. 2006

Computers & Mathematics with Applications

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The computation of the greatest common divisor (GCD) of many polynomials is a nongeneric problem. Techniques defining "approximate GCD" solutions have been defined, but the proper definition of the "approximate" GCD, and the way we can measure the strength of the approximation has remained open. This paper uses recent results on the representation of the GCD of many polynomials, in terms of factorisation of generalised resultants, to define the notion of "approximate GCD" and define the strength of any given approximation by solving an optimisation problem. The newly established framework is used to evaluate the performance of alternative procedures which have been used for defining approximate GCDs. (~)

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Generalized Resultant Theorem

Cited by 29 publications

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Gröbner bases and generalized sylvester matrices

Gröbner bases and generalized sylvester matrices

Solving Bezout-like polynomial equations for the design of interpolatory subdivision schemes

Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation

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